1.
Two philosophers alleged to deserve more credit than they have
received for their work in support of rigidity are Ruth Marcus and
Alvin Plantinga. Marcus' legacy has raised some clamor: the 1990s,
especially, saw “a historical controversy over the extent to
which the leading elements of Kripke's theory of reference and related
doctrines were derived from Marcus” (Garrett, 2005, p. 1600).
There is plenty of credit to go around, for the discovery of
rigidity's importance, but Marcus does hold the distinction of
clearing away by means of formal work important doubts about the
necessity of identity statements, and of distinguishing, in this
connection, names from descriptions: see the entry
intensional logic,
§2.4. On Plantinga's early role, see e.g., Davidson (2003, pp.
4, 15).

2.
One might expect terms for rigid designators to correspond in the
following way: a “strongly” rigid designator would be
obstinately rigid, referring to its object in all possible worlds. A
“weakly” rigid designator would be one that refers to its
object in just those possible worlds in which the object exists. It is
plausible to suppose that the question over whether statements like
‘Hesperus = Phosphorus’, or perhaps ‘If Hesperus
exists then Hesperus = Phosphorus’, are strongly necessary or
merely weakly necessary depends on whether ‘Hesperus’ and
‘Phosphorus’ are “strongly” rigid (obstinate)
or merely “weakly” rigid in this sense. However, as Kripke
has defined ‘strongly rigid’, ‘Hesperus’
cannot be a “strongly rigid designator”; that distinction
is reserved for designators that designate a necessarily existing
object (1980, pp. 48-9). Thus, given the truth of a classical
tradition according to which God and entities like numbers exist and
could not have failed to exist, ‘7’ or ‘God’
are “strongly rigid” in Kripke's sense: this is a special
case of obstinate rigidity.

3.
Of course there are skeptics who would say “pot-ā-to
– pot-ä-to,” just as there are in the alethic
parallel case (for a temporally-oriented skeptic, see Miller 2005).
Quinn, presented below as a critic of the counterpart reduction,
actually nuances his position to make room, in effect, for this idea
that there is no ontologically substantive difference expressed by
counterpart-talk, as opposed to rigidity-talk: he allows that
counterpart-talk could be presented as a mere verbal variant
of ordinary speech with rigid designators, in which case “Talk
of person-stages” would be “derivative from talk of
persons who persist through time” (Quinn 1978, pp.
353–354). But then there is really no point to the awkward
rendering of ordinary talk into the stages idiom. Quinn accordingly
directs his critical attention, in effect, to a metaphysically
substantive interpretation of counterpart-theoretic talk, in
criticizing reductive analyses. A further nuance, special to the
temporal as opposed to the alethic case, is that Quinn-inspired
reservations about verbal camouflage can arise without nonrigidity if
‘you’ is taken as a rigid designator but for a
sum of instantaneous parts, with respect to different times
within a possible world.

4.
Talk about an essence is not as common as talk about
the essence, but it can be helpful since many properties
apply to you and only you in all possible worlds: see Plantinga 1985,
pp. 85-7; 1974, p. 98. Context might select, so that “the”
relevant essence can differ, depending on whether the topic is
theoretical identity statements, say (see discussion in the main
entry), haecceitism or whatnot. Plantinga's definition of an essence
is given in the section on Individual Essences in the entry
actualism.

5.
Brentano, according to one interpretation anyway, maintained in
effect that we single out an individual rigidly in our minds by way of
grasping the individual's qualitative essence. “To have ideas of
distinct individuals,” in a transworld crowd of similar
singulars, “the difference between their properties must be
present to consciousness,” as Brown puts the idea (2000, p. 34):
so in order to explain successful singular thought, Brentano relies
upon an “exhaustive definition” that lists qualities
distinguishing the object of thought (Brentano, quoted in Brown 2000,
p. 35). Compare, by contrast, Plantinga (who is apparently unaware of
Brentano's connection with the offending view: 1974, pp. 94-95):

Why should we accept this idea? Suppose we consider an analogous
temporal situation. In Herbert Spiegelberg's book The
Phenomenological Movement there are pictures of Franz Brentano at
the age of 20 and of 70 respectively. The youthful Brentano looks much
like Apollo; the elderly Brentano resembles, instead, Jerome Hines in
his portrayal of the dying Czar in Boris Godounov. Most of us believe
that the same object exists at various distinct times; but do we know
of some empirically manifest property P such that a thing is
Brentano at a given time t if and only if it has P?
Surely not; and this casts no shadow whatever on the intelligibility
of our thought and talk about Brentano.

Despite the foregoing quotation and despite Plantinga’s abiding and
adamant support for transworld identity, Plantinga seems in later
writings to be unexpectedly non-committal with respect to whether
individuals last trans-temporally (2011, pp. 66–67). He
may favor a split position, insofar as temporal modality is concerned:
anyhow, his position seems broadly congenial with such a split (2011,
pp. 115, 119). As a dualist, he might hold that persons’
names like ‘Brentano’ are temporally rigid but
nevertheless not hold that names for physical objects are
also temporarily rigid (e.g., ‘the Empire State Building’,
‘K2’, or the name of a person’s body).

6.
Reduction of modal talk to talk about qualitative similarity is one
motive for invoking counterpart theory (Russell 2013, p. 88) and if
the counterparts construal is offered as a reduction (which is
typical) then the serious-counterpart theorist has a natural reply to
the charge that her interpretation of modal discourse camouflages a
change of meaning. She can reply that any such reduction is going to
have to involve some change in meaning in that it
clarifies and refines what natural language gropes to express but
which natural language in effect manages to express only incompletely
or with an admixture of confusion. Therefore if a criticism along the
relevant lines is to meet its mark, it has to go so far as to say that
counterpart theory fails to distill the message English
speakers convey and instead merely replaces it with a message
on a decidedly alien subject matter, using the same sentences.
Plantinga, of course, goes so far as to say just this. For him, the
counterpart theorist has modal opinions “clearly” at odds
with those of ordinary speakers, “hopes to remedy the situation
by giving the semantical reductive analysis in question,” and so
offers a reduction that hides the vast disagreement in opinion
(Plantinga 2003, pp. 222–223; cf, by contrast, Stalmaker 2003,
p. 186).

Plantinga himself seems doubtful about any reduction of modal
talk in terms of what could be expressed nonmodally: in order to
explain what he means by saying that objects exist with
respect to other possible worlds, say, he quickly appeals to
our original modal claims: so “To suppose that Socrates has
P in the actual world but lacks it in W is to suppose
only that Socrates does in fact have P but
would not have had it, had W been actual”
(1974, p. 92: emphasis added). Kripke denies that the reduction is
possible: “judgments involving directly expressed modal
locutions (‘it might have been the case that’) certainly
come earlier” than their articulation in terms of possible
worlds, and “in practice, no one who cannot understand”
these modal locutions properly would understand what it means, say, to
exist with respect to other possible worlds (1980, p. 19n).
Counterpart theory, on the other hand, promises the possibility of
reduction (but does not necessarily force it: Russell 2013, pp.
87–88).

A popular Quinean way of viewing reduction and conceptual change,
which would appeal to neither Plantinga nor Kripke but which will
worry many readers, threatens to weaken the foregoing position
articulated against counterpart theory. According to the Quinean
framework, there is no difference in principle between a proper
reduction of English modal sentences that preserves the original
content or that preserves what is good about the original content or
the like, on the one hand, and a change in subject matter that
redeploys the original sentences to express alien content, on the
other hand. Such a difference could only be a difference in degree,
such as that between tall and short, not a difference in kind, such as
that between apples and oranges. The foregoing position articulated
against counterpart theory could probably be recast in a Quinean
framework, albeit with some loss in strength; but there is reason to
suppose that reformulation is not needed anyway, because if we accept
any rigid designation in the first place, we have thereby already
committed ourselves against the Quinean framework. These issues of
conceptual change are discussed at length in LaPorte (2004, chaps 5
and 6, especially pp. 164–72).

By blurring the contrast between verbal and substantive differences,
Quineans would thereby blur the contrast between serious-counterpart
theorists and rigidity theorists. Still other philosophers would blur
the contrast between serious-counterpart theorists and rigidity
theorists even while maintaining the difference between matters verbal
and substantive: hence, it is sometimes suggested that that
ontological contrast itself is more verbal than substantive (see,
e.g., Sidelle 2002 p. 137). Compare, by contrast, Martí:
“any attempt to make rigidity and counterpart theory compatible
yields a notion that is, simply put, not the notion of rigidity”
(Martí 2003, p. 169; see also Sullivan 2005, pp. 583–587;
Torza 2013, pp. 744, 770).

7.
Of course, some expressions like ‘α’ may directly
refer even if ordinary proper names like ‘Petrarch’ do
not: so a descriptivist account of the semantics of ordinary names
that designates or “indexes” the world α might not
escape whatever problems of direct reference the descriptivist is
hoping to circumvent by way of descriptivism. Relatively
straightforward descriptivist accounts for the semantics of names that
follow the pattern of the α-indexing account cited in the text,
for its simplicity, may successfully capture the semantics of certain
names; but there are more sophisticated descriptivisms that seem to
have better prospects for broad application (for further discussion
and references, see LaPorte 2013, chapter 3 §2.4, chapter 8
§2.2). Similarly, there are other common ways to rigidify
descriptions besides world-indexing (e.g., one could use an operator
like Kaplan's ‘dthat’); and there are also ways other than
that of appealing to world-indexing, for motivating the case that
structured designators can be rigid de jure, even when the structured
designators are clearly not directly referential (or, again, clearly
not thoroughly directly referential: and again, for details and
references on the different options, the interested reader may turn
elsewhere: see LaPorte 2013, p. 63 n.12 and p. 140, respectively).

8.
Often an association between causal grounding and rigidity is
complicated by a further association between causal grounding and
indexicality. Thus, Putnam, for example, calls causally grounded terms
“indexical,” because they designate whatever has the
underlying essence of samples around the speaker.
‘Water’ and ‘whale’ are supposed to be
indexical; ‘hunter’ and ‘bachelor’ are not,
since they have analytic definitions. According to Putnam
“Kripke's doctrine that natural-kind words are rigid designators
and our doctrine that they are indexical are but two ways of making
the same point” (1975, p. 234). But these do not really seem to
be two ways of making the same point. For further citations and
discussion, see LaPorte 2000, §2; 2004, pp. 42-3.

9.
Reference is not secured by way of causal grounding in the relevant
respect, anyway: by way of ostension to an object in something like a
causal baptismal ceremony. It is a different question whether some
terms in the reference-fixing description are causally grounded. It is
hard to come up with descriptions free from such terms (Stanley 1997a,
p. 564; Devitt and Sterelny 1999, p. 60). If there are no such
descriptions available, then every rigid designator for a concrete
object may be said to be “broadly” causally
grounded in the respect that it is either grounded in the
primary way by means of ostension to an object in something like a
causal baptismal ceremony or it is hooked to a description
some terms in which are causally grounded in the primary way. In that
case, of course, broad causal grounding is ubiquitous and not
specially tied to rigidity: all singular concrete object designators,
including non-rigid definite descriptions, are broadly causally
grounded.

10.
Presumably, the relevant possibilities could not include all
metaphysically possible states of affairs: otherwise, it is hard to
see how (1) and (4) could share the same content, at least without
help from a sophisticated widescopism (Sosa shows how this could help
even though there are no modal operators: 2001, pp. 34-5n.7), which is
supposed to be a distinct suggestion.

12.
As suggested above, one might maintain a related line that the notion
of semantic content, assertion,
proposition, and so on must be reevaluated in light of the
distinction between assertoric content and ingredient sense: one might
say therefore that Kripke is onto one explication and that
assertoric content is yet another explication of the unrefined notion
semantic content, assertion, proposition,
and so on. Something like this position is adopted by Chalmers, who is
a pluralist about content (Chalmers 2006a, §1.4; what most
interests Chalmers is the division between what is epistemic
in Fregean sense and the modal phenomenon of rigidity: 2002, pp.
157-9). Even Dummett might tolerate the above proposal that Kripke is
onto one explication of semantic content,
assertion, proposition, and so on: he resists a
simple yes or no answer to the question whether ‘St. Joachim had
a daughter’ expresses the same proposition as ‘the father
of Mary had a daughter’: “The word
‘proposition’ is treacherous,” he cautions (p. 48).
Stanley (1997b, 132, 140, 155), by contrast, is much less favorably
disposed to say that Dummett's distinction could be said to bring to
light more than one notion of semantic content,
assertion, proposition, and the like.

13.
Further, we might hold that the theory of direct reference is merely
an empirical theory about natural language, and that even if it is
true, there are other possible languages in which a name spelled and
pronounced like ‘Petrarch’ is a disguised description
meaning the same as ‘the famous humanist most closely associated
in α with the Italian Renaissance’. For such a language,
rigidity does the work one would expect. So the work, even with
respect to names, is independent from the theory of direct reference
not only epistemically (for all many philosophers know
languages do not conform to the theory of direct reference and
rigidity performs its work anyway) but metaphysically (it is
metaphysically possible that languages do not conform to the
theory of direct reference and rigidity performs its work anyway).

14.
Other examples of the necessary a posteriori made famous by Kripke
may be accepted by direct reference theorists. Whether these examples
owe anything to rigidity may be contested (e.g., when the examples
concern kinds). Here it is enough to point out the vicinity and
general nature of the complications, which the reader can pursue.

15.
Thus, for Chalmers (2006b, §3.1) it is one of just a few
“core claims of two-dimensionalism,” as recent authors
understand it, that apriority obtains if and only if the primary
intension (indicated on the diagonal) is true at all scenarios.

16.
There are complications here: if the horizontal intension is a
“secondary intension,” in the specific way that Chalmers,
for one, understands it, then it is debatable whether the intension is
an infallible guide to whether a term is rigid. According to a
venerable tradition, there is an omniscient agent x, and it
is impossible that either x should have failed to be an
omniscient agent or that any being y such that y
≠ x should have been an omniscient agent. Something
similar can be said for ‘the simplest agent’. This makes
‘the omniscient agent’ and ‘the simplest
agent’ rigid designators. However, neither ‘the omniscient
agent’ nor ‘the simplest agent’ appears to designate
the same thing in all worlds along Chalmers' secondary intension. It
would appear that ‘the omniscient agent = the simplest
agent’ has a contingently true or false secondary
intension, even if tradition is right and the relevant
designators are actually rigid. But if tradition is right and these
designators are actually rigid, then ‘the omniscient
agent = the simplest agent’ is not contingently true or
false. Yet even if these complications afflict the specific version of
two-dimensionalism at hand, where that two-dimensionalism
does successfully capture the status of an identity
statement as necessary or contingent, it would appear to owe this to
rigidity: the horizontal intension successfully mirrors rigidity or
lack thereof in these cases.

17.
Does ‘watery stuff’ nonrigidly designate the property or
kind H2O, whether horizontally or diagonally, by virtue of
applying to stuff that is H2O with respect to some
worlds and not others? Or does ‘watery stuff’ rather fail
to designate the property H2O at all, though the expression
applies to stuff that is H2O, since it fails to
apply to stuff that is H2O with respect to some worlds?
There is no need here to answer these questions, which gesture toward
some of the complications with treating property terms as rigid.

18.
Possible replies to a general version of this objection are discussed
in
§2.2.3.3.
Of course, any or all of those replies might undermine a general
statement of the objection while failing to apply to this or that
specific version, with its particular context or nuances.

Notes to Supplement: Stipulating Identity Trans-world, Without Qualitative Criteria for a Designatum to Satisfy

1.
The problem is powerfully motivated by Augustine (see Matthews for
discussion: 2005, pp. 29–30), for example, whose best
illustration (‘walking’) reminds us that the problem
attends reference to properties and kinds, as well as individuals.
Augustine's example underscores the depth of the problem at hand: even
if we did know of essential criteria by which to apply a name
like yours and indeed even if the criteria were qualitative,
similar issues would arise with respect to how we establish reference
to the qualities themselves (a point Kripke observes: 1971,
p. 148).